2022 AIME II Problems/Problem 2
Problem
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability . When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability
. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is
, where
and
are relatively prime positive integers. Find
.
Solution 1
Let be Azar,
be Carl,
be Jon, and
be Sergey. The
circles represent the
players, and the arrow is from the winner to the loser with the winning probability on top.
This problem can be solved by using cases.
's opponent for the semifinals is
The probability 's opponent is
is
. Therefore the probability
wins the semifinals in this case is
. The other semifinal game is played between
and
, it doesn't matter who wins because
has the same probability of winning either one. The probability of
winning in the finals is
, so the probability of
winning the tournament is
's opponent for the semifinals is
/
It doesn't matter if 's opponent is
/
because
has the same probability of winning either one. The probability
's opponent is
/
is
. Therefore the probability
wins the semifinals in this case is
. The other semifinal game is played between
and
/
. In this case it matter who wins in the other semifinal game because the probability of
winning
or
/
is different.
's opponent for the finals is
For this to happen, must have won
/
in the semifinals
To be continued......
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=C14f91P2pYc
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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